Diploma thesis

a nonlinear, nonmagnetic material, that is free of charges and currents:
−∇ 2 � E + 1
c 2
∂2 ∂t2 � E = − 4π
c2 ∂2P� . (3.1)
∂t2 For dispersive nonlinear materials and three interacting waves, this equation can be
solved with a quasi-monochromatic Ansatz, and yields the intensity of the upconverted
field [14]:
I3(ω3) = (2π)524χ (2) 2
123 I1(ω1)I2(ω2)
n1n2n3λ2 3c L 2 sinc 2
� �
∆kL
. (3.2)
2
Equation 3.2 describes the coupling between the three involved fields. The efficiency
of SFG depends on the momentum mismatch ∆k, between the interacting waves,
where we assume strictly collinear waves propagation.
∆k = k3(ω3) − k2(ω2) − k1(ω1). (3.3)
Only if the combined momentum of all involved photons equals zero an efficient SFG
is possible. The Sellmeier equations or refractive indices for the crystal therefore have
to fulfil Eq. 3.4 under the restrictions of energy conservation (ω3 = ω2 + ω1).
n1ω1 + n2ω2 = n3ω3
(3.4)
The coupling constant of the three interacting waves with different polarizations,
in SFG, is given through the χ (2)
ijk-tensor. In lossless crystals, this tensor can be
reduced to a two dimensional matrix dij because of crystal symmetries [14].
⎛ ⎞
⎛
⎝
Px
Py
Pz
⎞
⎛
⎠ = ɛ0 ⎝
d11 d12 d13 d14 d15 d16
d21 d22 d23 d24 d25 d26
d31 d32 d33 d34 d35 d33
(Ex) 2
(Ey) 2
(Ez) 2
⎞ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎠ · ⎜ ⎟
⎜
⎜2EyEz
⎟
⎝2EzEx
⎠
2ExEy
(3.5)
The fields EiEj on the right hand side of equation 3.5 are the incoming pump fields.
They introduce a nonlinear polarization Pi, that emits a SFG field with the same
polarization, but a different frequency. In common nonlinear materials numerous
coupling constants dij are close to zero, and only a small number of possible threewave-interactions
remains. In the case of orthogonally polarized incoming pump
fields, this is called type-II SFG. Identically polarized pump waves are denoted by
type-I SFG.
Equation 3.2 still lacks the consideration of polychromatic waves, common for
pulsed laser sources. Hence, we generalize equation 3.2 to describe one broadband
laser pulses propagating through the crystal. The pulse exhibits a Gaussian shape
in the frequency domain, as follows:
6
I ′ (ω−ωp)2
−
2σ
p(ω) = Ipe 2 p (3.6)